Geometrical Tools for Quantum Euclidean Spaces

We apply one of the formalisms of noncommutative geometry to ℝ^N _q , the quantum space covariant under the quantum group SO _q (N). Over ℝ^N _q there are two SO _q (N)-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of ℝ³ _q . As in the case N=3, one has to slightly enlarge the algebra ℝ^N _q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over ℝ^N _q . While in our previous article the frame was found “by hand”, here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of ℝ^N _q with U _q so(N) into ℝ^N _q , an interesting result in itself.

Authors and Affiliations

1. Sektion Physik, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 München, Germany, , , , , , DE

   B. L. Cerchiai

2. Dip. di Matematica e Applicazioni, Fac. di Ingegneria, Università di Napoli, V. Claudio 21, 80125 Napoli, Italy, , , , , , IT

   G. Fiore

3. Laboratoire de Physique Théorique et Hautes Energies, Université de Paris-Sud, Bâtiment 211, 91405 Orsay, France, , , , , , FR

   J. Madore

Received: 4 March 2000 / Accepted: 11 October 2000

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